This document contains multiple thermodynamics problems involving phase equilibria, compressibility, expansivity, ideal gas behavior, and equations of state. Problem 1 involves determining the number of phase rule variables needed to specify the state of different multiphase systems. Problem 2 evaluates a claim about a reported quadruple point for an exotic substance. Problem 3 asks how a two-phase vapor/liquid system changes upon addition of pure ethanol.

1. Problem1:
How many phase rule variables must be specified to fix
the thermodynamic state of each of the following
systems?
(a) A sealed flask containing a liquid ethanol-water
mixture in equilibrium with its vapor.
(b) A sealed flask containing a liquid ethanol-water
mixture in equilibrium with its vapor and nitrogen.
(c) A sealed flask containing ethanol, toluene, and water
as two liquid phases plus vapor.

2. Problem 2:
A renowned laboratory reports quadruple-point coordinates
of 10.2 Mbar and 24.1°C for four-phase equilibrium of
allotropic solid forms of the exotic chemical β-miasmone.
Evaluate the claim.

3. Problem 3:
A system comprised of chloroform, 1,4-dioxane, and
ethanol exists as a two-phase vapor/liquid system at 50°
C and 55 kPa. After the addition of some pure ethanol,
the system can be returned to two-phase equilibrium at
the initial T and P. In what respect has the system
changed, and in what respect has it not changed?

4. Generally, volume expansivity β and isothermal
compressibility κ depend on T and P.
Prove that:
Problem 2

5. Problem 2
Express the volume expansivity and the isothermal compressibility
as functions of density ρ and its partial derivatives. For water at 50°
C and 1 bar, κ=44.18×10-6
bar-1
. To what pressure must water be
compressed at 50°C to change its density by 1%? Assume that κ is
independent of P.

6. The Tait equation for liquids is written for an isotherm as:
where V is molar or specific volume, V0 is the hypothetical molar
or specific volume at zero pressure, and A and B are positive
constants. Find an expression for the isothermal
compressibility consistent with this equation.
Problem 3

7. Problem 7:
For one of the substances in Table 3.2, compute
the final pressure when the substance is heated
from 15°C and 1 bar to 25°C at constant volume.

8. Problem 8:
A substance for which κ is a constant undergoes an
isothermal, mechanically reversible process from initial state
(P1, V1) to final state (P2, V2), where V is molar volume.
(a) Starting with the definition of κ, show that the path of the
process is described by:
(b) Determine an exact expression which gives the
isothermal work done on 1 mol of this constant-κ substance.

9. Problem 9: One mole of an ideal gas with CP = (7/2)R and CV =
(5/2)R expands from P1 = 8 bar and T1
= 600 K to P2
= 1 bar by
each of the following paths:
(a) Constant volume;
(b) Constant temperature;
(c) Adiabatically.
Assuming mechanical reversibility, calculate W, Q, ΔU, and ΔH
for each process. Sketch each path on a single PV diagram.

10. Problem 10.
An ideal gas, initially at 30°C and 100 kPa, undergoes the following cyclic processes in a closed system:
(a) In mechanically reversible processes, it is first compressed adiabatically to 500 kPa, then cooled at a constant
pressure of 500 kPa to 30°C, and finally expanded isothermally to its original state.
(b) The cycle traverses exactly the same changes of state, but each step is irreversible with an efficiency of 80%
compared with the corresponding mechanically reversible process. Note: The initial step can no longer be
adiabatic.
Calculate Q, W, ΔU, and ΔH for each step of the process and for the cycle. Take CP = (7/2)R and CV = (5/2)R.

12. Problem11:
One cubic meter of an ideal gas at 600 K and 1000 kPa
expands to five times its initial volume as follows:
(a) By a mechanically reversible, isothermal process.
(b) By a mechanically reversible, adiabatic process.
(c) By an adiabatic, irreversible process in which expansion is
against a restraining pressure of 100 kPa.
For each case calculate the final temperature, pressure, and
the work done by the gas. Take CP = 21 J・mol-1
K-1

15. Problem :
The vapor-phase molar volume of a particular compound is
reported as 23,000 cm3
・mol-1
at 300 K and 1 bar. No other data
are available. Without assuming ideal-gas behavior, determine
a reasonable estimate of the molar volume of the vapor at 300
K and 5 bar.

16. Problem
Estimate the following:
(a) The volume occupied by 18 kg of ethylene at 55°C and 35 bar.
(b) The mass of ethylene contained in a 0.25 m3 cylinder at 50°C and 115 bar.

17. Problem:To a good approximation, what is the molar
volume of ethanol vapor at 480°C and 6000 kPa? How
does this result compare with the ideal-gas value?

18. Problem: To what pressure does one fill a 0.15 m3
vessel
at 25°C in order to store 40 kg of ethylene in it?

19. Example: Comparison of Methods
Calculate the molar volume of ethylene at 40°C, 90 bar, using the
(a) ideal-gas law,
(b) the truncated virial equation, and
(c) the Pitzer correlation with the Lee-Kesler values for Z0
, Z1
Solution The critical parameters of ethylene are Pc
= 50.41 bar, Tc
= 282.34 K, ω = 0.087.
The reduced coordinates are (a) Ideal-gas law: The ideal-gas molar volume is
(b) Virial equation: We first calculate the second virial coefficient using following eqs.

20. The compressibility factor is calculated form the truncated virial equation
and the molar volume is

21. Lee-Kesler correlation The values of Z0
, Z1
, are